Conway's 1978 textbook *Functions of One Complex Variable I* gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner characterization is known. **Has any progress been made since then? And what simpler characterization is known today, if one is known?**

## Here is the problem definition:

An open connected set $G\subseteq \mathbb{C}$ is called a *Dirichlet Region* if for each continuous function $f:\partial_\infty G\rightarrow \mathbb{R}$ there is a continuous function $u:G^- \rightarrow \mathbb{R}$ such that $u$ is harmonic in $G$ and $u(z)=f(z)$ for all $z$ in $\partial_\infty G$.

(The notation $\partial_\infty G$ refers to the boundary of $G$ in $\mathbb{C}\cup\{\infty\}$, and $G^-$ denotes the closure of $G$ in $\mathbb{C}\cup\{\infty\}$.)

## The characterization given in the book is:

Given $a \in \partial_\infty G$, a *barrier* for $G$ at $a$ is a family $\{\psi_r: r>0\}$ of functions such that:

1. $\psi_r$ is well-defined and superharmonic on $B(a;r) \cap G$ with $0\leq \psi_r(z) \leq 1$

2. $\lim_{z\rightarrow a}\psi_r(z) = 0$, and

3. $\lim_{z\rightarrow w} \psi_r(z) = 1$ for $w$ in $G \cap \{w:|w-a|=r\}$.

An open connected set $G$ is a Dirichlet Region iff there is a barrier for $G$ at each point of $\partial_\infty G$.